How to remove/apply all modifiers of one object in python
Modifier removal

To remove a modifier from an object.

obj below is a reference to an object either obtained via 

obj = bpy.context.active_object
obj = bpy.context.object
# or
obj = bpy.data.objects[name]


If you want to remove a modifier it is done by name, not by type. You could have multiple array modifiers, but only want to remove
a specific one.

modifier_to_remove = obj.modifiers.get("Bevel")
obj.modifiers.remove(modifier_to_remove)

# or as a one-liner
obj.modifiers.remove(obj.modifiers.get("Bevel"))

# or remove all modifiers from an object
obj.modifiers.clear()


Modifier Apply

Using an operator  

As sambler pointed out

# operates on all selected objects (not just the scene's _active_object_)
bpy.ops.object.convert(target='MESH')


lower level  

If you want to apply all modifiers it's covered here: How to get a new mesh with modifiers applied using Blender Python API? - This involves using obj.to_mesh and there are a few things to consider, it: 


generates a new mesh from your current obj (leaves current obj.data intact) 
does not remove the modifiers automatically from the object. (unlike the Apply button on the Modifiers panel, or Convert to Mesh).


This snippet will:

"""
- make a reference to the current obj.data ( i call it old_mesh)
- make a new Mesh ( new_mesh) using the effect of the combined modifiers on `obj`
- remove all modifiers from `obj`
- assign the new Mesh to obj.data 
- remove old_mesh from the blend
"""


in code:

import bpy

context = bpy.context
scene = context.scene
obj = context.object

# get a reference to the current obj.data
old_mesh = obj.data

# settings for to_mesh
apply_modifiers = True
settings = 'PREVIEW'
new_mesh = obj.to_mesh(scene, apply_modifiers, settings)

# object will still have modifiers, remove them
obj.modifiers.clear()

# assign the new mesh to obj.data 
obj.data = new_mesh

# remove the old mesh from the .blend
bpy.data.meshes.remove(old_mesh)